Herman Cain’s beaming face graces the cover of the latest Newsweek, celebrated for his unlikely rise in the GOP presidential sweepstakes. But Mr. Cain may not be feeling so cheerful the day after the Las Vegas debate, where he faced a withering assault on his signature 9-9-9 tax plan.

And right after Tuesday’s debate, Cain was forced to admit to host Anderson Cooper of CNN that he had misspoken in an earlier interview about being willing to release all the prisoners from Guantánamo Bay prison camp in a hypothetical deal with Al Qaeda.

Cain is a novice politician. That hasn’t prevented him from rising in the polls to a virtual tie with former Massachusetts Gov. Mitt Romney for Republican front-runner. In fact, at a time of disgust with politicians and Washington, Cain’s nonpolitical background is a plus with many voters.

But there are times when a lack of experience shows, and he may have a hard time solidifying his new standing as a top-tier candidate. Surely, Cain anticipated criticism of his 9-9-9 plan during the debate, but he struggled to beat back the attacks on his idea.

Mr. Cooper noted that prominent conservatives have said the plan would raise taxes on middle-class and lower-income voters. Former Sen. Rick Santorum of Pennsylvania asserted that 84 percent of Americans would pay higher taxes under 9-9-9. The plan would replace all current federal taxes with a 9 percent business tax, 9 percent personal income tax, and a 9 percent federal sales tax.

Cain dismissed the critiques as “simply not true,” without providing much detail, and then suggested people read the campaign-approved analysis by the firm Fiscal Associates at his website, HermanCain.com.

Minnesota Rep. Michele Bachmann said the plan effectively institutes a value-added tax, which taxes every stage of production and is dangerous because it’s “hidden,” she says. Texas Rep. Ron Paul complained that the 9-9-9 plan raises revenues and is “regressive.” Other candidates went after the implications of a new federal sales tax.

“Herman, I love you, brother, but let me tell you something: You don’t have to have a big analysis to figure this thing out,” said Texas Gov. Rick Perry. “Go to New Hampshire, where they don’t have a sales tax, and you’re fixin’ to give ‘em one.”

Governor Perry also pointed out that in Nevada, which already has a state sales tax, residents would be paying an additional 9 percent on purchases. Mr. Romney jumped on the same point. Cain complained both men were mixing up their fruits.

“The state tax is an apple,” Cain said. “We are replacing the current tax code with oranges. So it’s not correct to mix apples and oranges.”

Some debate viewers may well have come away agreeing with former House Speaker Newt Gingrich’s assessment. He praised Cain for having the courage to propose a “very big idea,” but concluded that “there are much more complexities than Herman lets on.”

Foreign policy has been a particular Achilles’ heel for Cain. Before the Tuesday debate, CNN’s Wolf Blitzer asked Cain if he would trade all the prisoners in the US prison camp at Guantanamo Bay in a theoretical deal with Al Qaeda to release a US soldier.

Cain’s answer: "I could see myself authorizing that kind of transfer, but what I would do is I would make sure that I got all of the information, I got all of the input, considered all of the options, and then, the president has to make a judgment call.”

During the debate, Mr. Cooper brought up that statement, and Cain said he would have a policy of no negotiation with terrorists. He added that he did not recall being asked about such a negotiation with Al Qaeda. Cooper pointed out that that was in fact the case.

After the debate, Cain said he mixed up the hypothetical case Mr. Blitzer had presented with the real example from earlier Tuesday, when Israel released more than 1,000 Palestinian prisoners to free Israeli soldier Gilad Shalit.

"Here is how I was misspeaking about our situation," Cain told Cooper. "I said Prime Minister Benjamin Netanyahu had a lot of things to consider in order to make that decision. What I'm saying is I can't say he did the right thing or the wrong thing just based on the numbers. That is the point I was trying to make. I probably misspoke when I went from that situation over to this situation. You got to have all the facts."

It remains to be seen how Cain will fare in polls going forward. But for now, his fans are legion. On Tuesday, a new "super political-action committee" called the 999 PAC announced its debut. It is organized by Republican strategist Jordan Gehrke and plans to hire staff to help Cain in the early nominating states of Iowa, New Hampshire, Nevada, South Carolina, Michigan, Arizona, and Florida.